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doc: Periodic grammarcheck of documentation. * doc/interpreter/basics.txi, doc/interpreter/container.txi, doc/interpreter/contrib.txi, doc/interpreter/diagperm.txi, doc/interpreter/errors.txi, doc/interpreter/install.txi, doc/interpreter/sparse.txi, libinterp/corefcn/ellipj.cc, libinterp/corefcn/mappers.cc, libinterp/corefcn/regexp.cc, libinterp/corefcn/tril.cc, libinterp/dldfcn/__init_fltk__.cc, libinterp/dldfcn/fftw.cc, libinterp/interpfcn/dirfns.cc, libinterp/interpfcn/input.cc, libinterp/octave-value/ov-fcn-inline.cc, libinterp/octave-value/ov-struct.cc, scripts/@ftp/cd.m, scripts/general/interp1.m, scripts/general/num2str.m, scripts/image/ind2rgb.m, scripts/image/rgb2ind.m, scripts/io/importdata.m, scripts/io/textread.m, scripts/java/javamem.m, scripts/linear-algebra/condest.m, scripts/linear-algebra/onenormest.m, scripts/miscellaneous/error_ids.m, scripts/miscellaneous/getfield.m, scripts/miscellaneous/setfield.m, scripts/plot/area.m, scripts/plot/pcolor.m, scripts/plot/stairs.m, scripts/set/powerset.m, scripts/sparse/bicg.m, scripts/sparse/bicgstab.m, scripts/sparse/cgs.m, scripts/specfun/ellipke.m, scripts/special-matrix/gallery.m, scripts/strings/strjoin.m, scripts/strings/strsplit.m, scripts/testfun/__have_feature__.m, scripts/testfun/__printf_assert__.m, scripts/testfun/__prog_output_assert__.m, scripts/testfun/__run_test_suite__.m: grammarcheck documentation.
author Rik <rik@octave.org>
date Sat, 22 Jun 2013 19:47:32 -0700
parents e2de3c8882be
children e39f00a32dc7
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1 ## Copyright (C) 2001 David Billinghurst
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2 ## Copyright (C) 2001 Paul Kienzle
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3 ## Copyright (C) 2003 Jaakko Ruohio
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4 ##
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5 ## This file is part of Octave.
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6 ##
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7 ## Octave is free software; you can redistribute it and/or modify it
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8 ## under the terms of the GNU General Public License as published by
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9 ## the Free Software Foundation; either version 3 of the License, or (at
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10 ## your option) any later version.
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11 ##
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12 ## Octave is distributed in the hope that it will be useful, but
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13 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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14 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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15 ## General Public License for more details.
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16 ##
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17 ## You should have received a copy of the GNU General Public License
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18 ## along with Octave; see the file COPYING. If not, see
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19 ## <http://www.gnu.org/licenses/>.
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20
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21 ## -*- texinfo -*-
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22 ## @deftypefn {Function File} {} ellipke (@var{m})
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23 ## @deftypefnx {Function File} {} ellipke (@var{m}, @var{tol})
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24 ## @deftypefnx {Function File} {[@var{k}, @var{e}] =} ellipke (@dots{})
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25 ## Compute complete elliptic integral of the first K(@var{m}) and second
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26 ## E(@var{m}) kind.
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27 ##
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28 ## @var{m} is either real array or scalar with 0 @leq{} m @leq{} 1.
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29 ##
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30 ## @var{tol} is currently ignored (@sc{matlab} uses this to allow faster,
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31 ## less accurate approximation).
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32 ##
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33 ## Ref: Abramowitz, Milton and Stegun, Irene A. Handbook of Mathematical
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34 ## Functions, Dover, 1965, Chapter 17.
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35 ## @seealso{ellipj}
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36 ## @end deftypefn
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37
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38 ## Author: David Billinghurst <David.Billinghurst@riotinto.com>
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39 ## Author: Paul Kienzle <pkienzle@users.sf.net>
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40 ## Author: Jaakko Ruohio
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41
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42 function [k, e] = ellipke (m)
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43
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44 if (nargin < 1 || nargin > 2)
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45 print_usage ();
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46 endif
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47
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48 k = e = zeros (size (m));
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49 m = m(:);
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50 if any (!isreal (m))
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51 error ("ellipke must have real m");
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52 endif
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53 if any (m > 1)
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54 error ("ellipke must have m <= 1");
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55 endif
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56
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57 Nmax = 16;
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58 idx = find (m == 1);
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59 if (!isempty (idx))
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60 k(idx) = Inf;
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61 e(idx) = 1;
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62 endif
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63
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64 idx = find (m == -Inf);
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65 if (!isempty (idx))
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66 k(idx) = 0;
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67 e(idx) = Inf;
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68 endif
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69
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70 ## Arithmetic-Geometric Mean (AGM) algorithm
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71 ## ( Abramowitz and Stegun, Section 17.6 )
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72 idx = find (m != 1 & m != -Inf);
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73 if (!isempty (idx))
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74 idx_neg = find (m < 0 & m != -Inf);
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75 mult_k = 1./sqrt (1 - m(idx_neg));
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76 mult_e = sqrt (1 - m(idx_neg));
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77 m(idx_neg) = -m(idx_neg)./(1 - m(idx_neg));
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78 a = ones (length (idx), 1);
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79 b = sqrt (1 - m(idx));
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80 c = sqrt (m(idx));
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81 f = 0.5;
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82 sum = f*c.*c;
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83 for n = 2:Nmax
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84 t = (a + b)/2;
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85 c = (a - b)/2;
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86 b = sqrt (a.*b);
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87 a = t;
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88 f = f * 2;
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89 sum = sum + f*c.*c;
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90 if (all (c./a < eps)) break; endif
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91 endfor
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92 if (n >= Nmax) error ("ellipke: not enough workspace"); endif
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93 k(idx) = 0.5*pi./a;
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94 e(idx) = 0.5*pi.*(1 - sum)./a;
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95 k(idx_neg) = mult_k.*k(idx_neg);
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96 e(idx_neg) = mult_e.*e(idx_neg);
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97 endif
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98
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99 endfunction
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100
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101 %% Test complete elliptic functions of first and second kind
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102 %% against "exact" solution from Mathematica 3.0
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103 %!test
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104 %! m = [0.0; 0.01; 0.1; 0.5; 0.9; 0.99; 1.0 ];
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105 %! [k,e] = ellipke (m);
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106 %!
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107 %! # K(1.0) is really infinity - see below
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108 %! k_exp = [1.5707963267948966192;
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109 %! 1.5747455615173559527;
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110 %! 1.6124413487202193982;
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111 %! 1.8540746773013719184;
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112 %! 2.5780921133481731882;
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113 %! 3.6956373629898746778;
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114 %! 0.0 ];
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115 %! e_exp = [1.5707963267948966192;
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116 %! 1.5668619420216682912;
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117 %! 1.5307576368977632025;
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118 %! 1.3506438810476755025;
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119 %! 1.1047747327040733261;
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120 %! 1.0159935450252239356;
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121 %! 1.0 ];
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122 %! if k(7)==Inf, k(7)=0; endif;
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123 %! assert (k, k_exp, 8*eps);
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124 %! assert (e, e_exp, 8*eps);
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125
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126 %% Test against A&S Table 17.1
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127 %!test
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128 %! m = [0:5:50]'/100;
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129 %! k_exp = [1.570796326794897;
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130 %! 1.591003453790792;
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131 %! 1.612441348720219;
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132 %! 1.635256732264580;
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133 %! 1.659623598610528;
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134 %! 1.685750354812596;
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135 %! 1.713889448178791;
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136 %! 1.744350597225613;
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137 %! 1.777519371491253;
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Mike Miller <mtmiller@ieee.org>
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138 %! 1.813883936816983;
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139 %! 1.854074677301372 ];
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140 %! e_exp = [1.570796327;
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141 %! 1.550973352;
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142 %! 1.530757637;
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143 %! 1.510121831;
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144 %! 1.489035058;
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Mike Miller <mtmiller@ieee.org>
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145 %! 1.467462209;
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Mike Miller <mtmiller@ieee.org>
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146 %! 1.445363064;
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Mike Miller <mtmiller@ieee.org>
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147 %! 1.422691133;
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148 %! 1.399392139;
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149 %! 1.375401972;
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150 %! 1.350643881 ];
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151 %! [k,e] = ellipke (m);
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152 %! assert (k, k_exp, 1e-15);
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153 %! assert (e, e_exp, 1e-8);
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154
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155 %% Test input validation
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156 %!error ellipke ()
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157 %!error ellipke (1,2,3)